On the minimal polynomial of Gauss periods for prime powers

For a positive integer $m$, set $\zeta_{m}=\exp(2\pi i/m)$ and let ${\bf Z}_{m}^{*}$ denote the group of reduced residues modulo $m$. Fix a congruence group $H$ of conductor $m$ and of order $f$. Choose integers $t_{1},\dots,t_{e}$ to represent the $e=\phi(m)/f$ cosets of $H$ in ${\bf Z}_{m}^{*}$. The Gauss periods

$\displaylines{ \theta_{j} =\sum_{x \in H} \zeta_{m}^{t_{j}x} \;\;\; (1 \leq j \leq e) }$

corresponding to $H$ are conjugate and distinct over ${\bf Q}$ with minimal polynomial

$\displaylines{ g(x) = x^{e} + c_{1}x^{e-1} + \cdots + c_{e-1} x + c_{e}. }$

To determine the coefficients of the period polynomial $g(x)$ (or equivalently, its reciprocal polynomial $G(X)=X^{e}g(X^{-1}))$ is a classical problem dating back to Gauss. Previous work of the author, and Gupta and Zagier, primarily treated the case $m=p$, an odd prime, with $f >1$ fixed. In this setting, it is known for certain integral power series $A(X)$ and $B(X)$, that for any positive integer $N$

$\displaylines{ G(X) \equiv A(X)\cdot B(X)^{\frac{p-1}{f}} \;\;\;({\rm mod}\;X^{N}) }$

holds in ${\bf Z}[X]$ for all primes $p \equiv 1({\rm mod}\; f)$ except those in an effectively determinable finite set. Here we describe an analogous result for the case $m=p^{\alpha}$, a prime power ( $\alpha > 1$). The methods extend for odd prime powers $p^{\alpha}$ to give a similar result for certain twisted Gauss periods of the form

$\displaylines{ \psi_{j} = i^{*} \sqrt{p} \sum_{x \in H} (\frac{t_{j}x}{p}) \zeta_{p^{\alpha}}^{t_{j}x} \;\;(1 \leq j \leq e),}$

where $(\frac{ }{p})$ denotes the usual Legendre symbol and $i^{*}= i^{\frac{(p-1)^{2}}{4}}$.